# Continuity argument

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A **continuity argument** relies on the basic fact that any subset of a connected set which is simultaneously closed, open, and non-empty must be the whole set. In applications to nonlinear evolution equations, the connected set in question is simply the time axis (or some interval in this axis), although one can also use higher-dimensional connected sets, such as a ball in some normed vector space.

The continuity method combines particularly well with a bootstrap argument.